# dimensionless

In dimensional analysis, a dimensionless quantity (or more precisely, a quantity with the dimensions of 1) is a quantity without any physical units and thus a pure number. Such a number is typically defined as a product or ratio of quantities which do have units, in such a way that all units cancel.

## Examples

"out of every 10 apples I gather, 1 is rotten." -- the rotten-to-gathered ratio is (1 apple) / (10 apples) = 0.1 = 10%, which is a dimensionless quantity. Another more typical example in physics and engineering is the measure of plane angles with the unit of "radian". An angle measured this way is expressed as the ratio of the length of an arc lying on a circle (with its center being the vertex of the angle) swept out by the angle to the length of the radius of the circle. The ratio (length divided by length) is dimensionless.

Dimensionless quantities are widely used in the fields of mathematics, physics, engineering, and economics but also in everyday life. Whenever one measures any physical quantity, they are measuring that physical quantity against a like dimensioned standard. Whenever one commonly measures a length with a ruler or tape measure, they are counting tick marks on the standard of length they are using, which is a dimensionless number. When they attach that dimensionless number (the number of tick marks) to the units that the standard represents, they conceptually are referring to a dimensionful quantity. A quantity Q is defined as the product of that dimensionless number n (the number of tick marks) and the unit U (the standard):
:
But, ultimately, people always work with dimensionless numbers in reading measuring instruments and manipulating (changing or calculating with) even dimensionful quantities.

In case of dimensionless quantities the unit U is a quotient of like dimensioned quantities that can be reduced to a number (kg/kg = 1, μg/g = 1e-6). Dimensionless quantities can also carry dimensionless units like % (=0.01), ppt (=1e-3), ppm (=1e-6), ppb (=1e-9).

The CIPM Consultative Committee for Units toyed with the idea of defining the unit of 1 as the 'uno', but the idea was dropped. [1] [2] [3] [4]

## Properties

• A dimensionless quantity has no physical unit associated with it. However, it is sometimes helpful to use the same units in both the numerator and denominator, such as kg/kg, to show the quantity being measured.
• A dimensionless proportion has the same value regardless of the measurement units used to calculate it. It has the same value whether it was calculated using the SI system of units or the imperial system of units. This doesn't hold for all dimensionless quantities; it is guaranteed to hold only for proportions.

## Buckingham π-theorem

According to the Buckingham π-theorem of dimensional analysis, the functional dependence between a certain number (e.g., n) of variables can be reduced by the number (e.g., k) of independent dimensions occurring in those variables to give a set of p = nk independent, dimensionless quantity. For the purposes of the experimenter, different systems which share the same description by dimensionless quantity are equivalent.

### Example

The power consumption of a stirrer with a particular geometry is a function of the density and the viscosity of the fluid to be stirred, the size of the stirrer given by its diameter, and the speed of the stirrer. Therefore, we have n = 5 variables representing our example.

Those n = 5 variables are built up from k = 3 dimensions which are:
• Length: L (m)
• Time: T (s)
• Mass: M (kg)
According to the π-theorem, the n = 5 variables can be reduced by the k = 3 dimensions to form p = nk = 5 − 3 = 2 independent dimensionless numbers which are in case of the stirrer
• Reynolds number (This is the most important dimensionless number; it describes the fluid flow regime)
• Power number (describes the stirrer and also involves the density of the fluid)

## List of dimensionless quantities

There are infinitely many dimensionless quantities and they are often called numbers. Some of those that are used most often have been given names, as in the following list of examples (alphabetical order):
Name Field of application
Abbe numberoptics (dispersion in optical materials)
Albedoclimatology, astronomy (reflectivity of surfaces or bodies)
Archimedes numbermotion of fluids due to density differences
Bagnold numberflow of grain, sand, etc. [5]
Biot numbersurface vs. volume conductivity of solids
Bodenstein numberresidence-time distribution
Bond numbercapillary action driven by buoyancy [6]
Brinkman numberheat transfer by conduction from the wall to a viscous fluid
Brownell Katz numbercombination of capillary number and Bond number
Capillary numberfluid flow influenced by surface tension
Coefficient of static frictionfriction of solid bodies at rest
Coefficient of kinetic frictionfriction of solid bodies in translational motion
Colburn j factordimensionless heat transfer coefficient
Courant-Friedrich-Levy number  non-hydrostatic dynamics [7]
Damköhler numbersreaction time scales vs. transport phenomena
Darcy friction factorfluid flow
Dean numbervortices in curved ducts
Deborah numberrheology of viscoelastic fluids
Drag coefficientflow resistance
Eckert numberconvective heat transfer
Ekman numbergeophysics (frictional (viscous) forces)
Elasticity (economics)widely used to measure how demand or supply responds to price changes
Eötvös numberdetermination of bubble/drop shape
Euler numberhydrodynamics (pressure forces vs. inertia forces)
Fanning friction factorfluid flow in pipes [8]
Feigenbaum constantschaos theory (period doubling) [9]
Fine structure constantquantum electrodynamics (QED)
Foppl–von Karman numberthin-shell buckling
Fourier numberheat transfer
Fresnel numberslit diffraction [10]
Froude numberwave and surface behaviour
Gainelectronics (signal output to signal input)
Galilei numbergravity-driven viscous flow
Graetz numberheat flow
Grashof numberfree convection
Hagen numberforced convection
Karlovitz numberturbulent combustion
Knudsen numbercontinuum approximation in fluids
Kt/Vmedicine
Laplace numberfree convection within immiscible fluids
Lewis numberratio of mass diffusivity and thermal diffusivity
Lockhart-Martinelli parameterflow of wet gases [11]
Lift coefficientlift available from an airfoil at a given angle of attack
Mach numbergas dynamics
Magnetic Reynolds numbermagnetohydrodynamics
Manning roughness coefficientopen channel flow (flow driven by gravity) [12]PDF (109 KiB)
Marangoni numberMarangoni flow due to thermal surface tension deviations
Morton numberdetermination of bubble/drop shape
Nusselt numberheat transfer with forced convection
Ohnesorge numberatomization of liquids, Marangoni flow
Peel numberadhesion of microstructures with substrate [13]
Pimathematics (ratio of a circle's circumference to its diameter)
Poisson's ratioelasticity (load in transverse and longitudinal direction)
Power factorelectronics (real power to apparent power)
Power numberpower consumption by agitators
Prandtl numberforced and free convection
Pressure coefficientpressure experienced at a point on an airfoil
Rayleigh numberbuoyancy and viscous forces in free convection
Refractive indexelectromagnetism, optics
Reynolds numberflow behavior (inertia vs. viscosity)
Richardson numbereffect of buoyancy on flow stability [14]
Rockwell scalemechanical hardness
Rossby numberinertial forces in geophysics
Schmidt numberfluid dynamics (mass transfer and diffusion) [15]
Sherwood numbermass transfer with forced convection
Sommerfeld numberboundary lubrication [16]
Stanton numberheat transfer in forced convection
Stefan numberheat transfer during phase change
Stokes numberparticle dynamics
Strainmaterials science, elasticity
Strouhal numbercontinuous and pulsating flow [17]
Taylor numberrotating fluid flows
van 't Hoff factorquantitative analysis (Kf and Kb)
Weaver flame speed numberlaminar burning velocity relative to hydrogen gas [18]
Weber numbermultiphase flow with strongly curved surfaces
Weissenberg numberviscoelastic flows [19]
Womersley numbercontinuous and pulsating flows [20]

## Dimensionless physical constants

Certain physical constants, such as the speed of light in a vacuum, are normalized to 1 if the units for time, length, mass, charge, and temperature are chosen appropriately. The resulting system of units is known as Planck units. However, a handful of dimensionless physical constants cannot be eliminated in any system of units; their values must be determined experimentally. The resulting fundamental physical constants include:

Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities.
Quantity is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with quality, substance, change, and relation. Quantity was first introduced as quantum, an entity having quantity.
units of measurement have played a crucial role in human endeavour from early ages up to this day. Disparate systems of measurement used to be very common. Now there is a global standard, the International System (SI) of units, the modern form of the metric system.
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied.
This article or section is in need of attention from an expert on the subject.
Quantity is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with quality, substance, change, and relation. Quantity was first introduced as quantum, an entity having quantity.
radian, in mathematics, is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees. It is represented by the symbol "rad" or, more rarely, by the superscript c (for "circular measure"). For example, an angle of 1.2 radians would be written as "1.
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge.
Engineering is the applied science of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. The American Engineers' Council for Professional Development, also known as ECPD,[1] (later ABET [2]
Economics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Greek for oikos (house) and nomos (custom or law), hence "rules of the house(hold).
worldwide view of the subject.
Metrology (from Greek 'metron' (measure), and 'logos' (study of)) is the science of measurement.
variable (IPA pronunciation: [ˈvæɹiəbl]) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression.
In mathematics, an independent variable is any of the arguments, i.e. "inputs", to a function. These are contrasted with the dependent variable, which is the value, i.e. the "output", of the function.
dimension (Latin, "measured out") is a parameter or measurement required to define the characteristics of an object—i.e., length, width, and height or size and shape.
Quantity is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with quality, substance, change, and relation. Quantity was first introduced as quantum, an entity having quantity.
Quantity is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with quality, substance, change, and relation. Quantity was first introduced as quantum, an entity having quantity.
Electric power is defined as the rate at which electrical energy is transferred by an electric circuit. The SI unit of power is the watt.

When electric current flows in a circuit with resistance, it does work.
magnetic stirrer is a type of laboratory equipment consisting of a rotating magnet or stationary electomagnets creating a rotating magnetic field. The stirrer is used to cause a stir bar, immersed in a liquid to be stirred, to spin very quickly, stirring it.
In physics, density is mass m per unit volume V—how heavy something is compared to its size. A small, heavy object, such as a rock or a lump of lead, is denser than a lighter object of the same size or a larger object of the same weight, such as pieces of
Viscosity is a measure of the resistance of a fluid to deform under either shear stress or extensional stress. It is commonly perceived as "thickness", or resistance to flow.
diameter (Greek words diairo = divide and metro = measure) of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle.
Speed is the rate of motion, or equivalently the rate of change in position, many times expressed as distance d traveled per unit of time t.

Speed is a scalar quantity with dimensions distance/time; the equivalent vector quantity to speed is known as
In fluid mechanics, the Reynolds number is the ratio of inertial forces (vsρ) to viscous forces (μ/L) and consequently it quantifies the relative importance of these two types of forces for given flow conditions.
The power number Np (also known as Newton number) is a dimensionless number relating the resistance force to the inertia force. In engineering, this number, along with the Reynolds number, is one of the most widely employed dimensionless numbers.